Abstract

The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence τ1-Rϕ(β)-τ2, in which Rϕ(β) is a rotation by an angle β around an axis with phase ϕ in the transverse plane and τ1 and τ2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for τ1 and τ2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr-Purcell-Meiboom-Gill sequences identifies an eight-pulse phase alternating scheme that generates a propagator nearly equal to the identity operator.

Euler angles (a) α(τp), (b) β(τp), and (c) γ(τp) for a EBURP-2 pulse are shown as a function of the ratio of resonance offset and B1 field, Ω/ω1. (red) The limiting near-resonance linear dependence of α(τp) over the bandwidth ± 0.3Ω/ω1 is shown.

Minimum-energy broadband excitation pulse. (a) The (black) ωx(t) and (red) ωy(t) components of the pulse shape are shown. (b) The excitation profile for z-magnetization is shown as a function of the resonance offset Ω. (c) The Euler angles (black) α(τp) and (red) γ(τp) and (d) β(τp) are shown as a function of the resonance offset Ω. The pulse had a length of 100 μs and ω1/(2π) = 20 kHz.

Refocused transverse magnetization as a function of the ratio of resonance offset and B1 field, Ω/ω1. Plots show refocused (b,d,f) x-magnetization or (a,c,e) y-magnetization following CPMG pulse trains beginning with initial (b,d,f) x-magnetization or (a,c,e) y-magnetization. Panels (a) and (b) show the refocused magnetization after one cycle of CPMG pulse train. Panels (c,d,e,f) show the refocused magnetization after n cycles, averaged over n = 1 to 100. For panels (e) and (f), 10% rf inhomogeneity (FWHM) was included in the calculation. Each cycle consists of 8 pulses for all four schemes. The color coding is (blue) conventional x-phase, (black) Yip and Zuiderweg phase alternating scheme, (red) proposed eight-pulse phase alternating scheme, and (green) XY-8 scheme. All calculations used ω1/2π = 2500 Hz, τcp = 1.25 ms, in which τcp is the spacing between π pulses. The calculations were performed using the software package SIMPSON [].

CPMG performance demonstrated from normalized intensities for the 1H resonance of H2O in a doped water sample. (a) Refocused x-magnetization for the (black) proposed and (red) Yip and Zuiderweg phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains, corresponding to 1, 5, and 10 repetitions of an 8-pulse CPMG block, as described in Experimental Methods. (b) Refocused (black) x-magnetization and (red) y-magnetization for the proposed eight-pulse phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains. (c–e) Refocused x-magnetization for (red, circle) the proposed eight-pulse scheme and (green, square) the XY-8 scheme for (c) 8 ms, (d) 40 ms, and (e) 80 ms pulse trains overlaid onto the x-magnetization from numerical simulations using (black) the proposed scheme and (blue) the XY-8 scheme. 10% rf inhomogeneity (FWHM) was included in the calculations. The magnetizations in panels (c)–(d) were normalized to the values at zero resonance offset for clarity. The relative performance of the proposed and XY-8 schemes for y-magnetization are similar to the performance for x-magnetization and is not shown. The Yip and Zuiderweg scheme does not refocus orthogonal y-phase magnetization and results are not shown for this sequence.

The x-, y- and z-components of effective Hamiltonians from EEHT calculations for (a,b,c) the XY-8 and (d,e,f) proposed 8-pulse schemes, multiplied by the cycle time, τc. The product reflects the residual x-, y- and z-rotations each cycle. The calculations were performed at τcp/τ180 values of (red) 0, (green)1, (blue) 10 and (magenta) 100.